THE DIRECT KINEMATICS OF REDUNDANT PARALLEL ROBOT FOR PREDICTIVE CONTROL

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1 HE DIREC KINEMAICS OF REDUNDAN PARALLEL ROBO FOR PREDICIVE CONROL BELDA KVĚOSLAV, BÖHM JOSEF, VALÁŠEK MICHAEL Department of Adaptive Systems, Institute of Information heory and Automation, Academy of Sciences of the Czech Repubic, Pod vodárenskou věží, 8 8 Praha 8 - Libeň, fax: , emais: beda@utia.cas.cz, bohm@utia.cas.cz. Abstract: he parae robots seem to be one of the promising ways to improve accuracy and speed. At their deveopment some new probems appear. his paper deas with the design of direct kinematics for rea-time path contro of panar redundant parae robots. he main reason for its using is a fact that the direct kinematics gives possibiity to use Cartesian coordinates as against joint ones and by this consideraby simpifies mode of the robot and consecutivey computation of high eve contro based on knowedge of such mode, eg. Predictive Contro. As a subtask of design of the direct kinematics, the trajectory panning is discussed. Key-Words: Direct kinematics, Panar redundant parae robot, Goba operationa-space contro, DAE-ODE equations, Panning of the rajectory. INRODUCION he most topica industria robots and manipuators do not cope with increasing requirements on speed and accuracy. herefore, new approaches of their construction are being found. Parae robots, especiay redundanty actuated, seem to be one of the promising ways to sove these requirements []. And, moreover, they have severa advantages over traditiona seria robots. he main is the foowing: A or amost a drives are ocated on the basic frame and truss construction of the robot eads to higher stiffness than in seria types. It is advantageous for accurate machining and positioning. On the other hand the parae robots have one constrain: hat is given in more possibiity of arms coision. But this can be soved if this constraint woud be taken into account at the panning of desired trajectory. However, this disadvantage does not markedy keep down movement of the robot. As an exampe, et us consider one such redundanty actuated panar parae robot (Figure ). It consists of the basic frame, which at the same time encoses workspace of the robot, four independent drives, movabe patform and eight arms, which connect the movabe patform with the basic frame. he arms are paraey situated. From the mechanica point of view, this robot has one drive and one pair of arms redundant, because generay the number of degrees of freedom of body in a pane is ony three. Accordingy, for contro of the robot and for its mechanica determination, ony three pairs of appropriate arms are necessary. But in this case, the singuar position in the workspace wi appear. herefore the redundant drive is used in order to overcome the probem. And, moreover, it improves stiffness and rotation speed of movabe patform and gives the possibiity to compy with the other additiona contro requirements. he aim of this paper is investigation of the direct kinematics for rea-time contro of redundant parae structure of the robot at using of the speciay panned trajectory. R -

2 ψ ϕ B A A 3 ψ B A C C 3 E C C ϕ 3 ϕ ϕ B ψ ψ A B 3 ψ 3 Figure - Scheme of panar parae robot with the most important geometrica description (he coordinates of center E of movabe patform and its ange of winding ψ). PROBLEM FORMULAION From the high-eve contro (e.g. Predictive Contro [] ) design point of view, the principa task is a choice of suitabe mode of the robot. On it the design of contro depends. he robot-manipuator is a mutibody system, which can be described by Lagrange s equations of mixed type. hese equations ead to the differentia-agebraic equations (DAE) in the foowing form: Ms Φ s λ = g + u () f(s( t)) = where M is a mass matrix, s is a vector of physica coordinates (their number is higher than the number of degrees of freedom), Φs is a Jacobian of the system, λ are Lagrange s mutipiers, g is a vector of right sides, matrix transforms the inputs u (four torques) into four drives and f(s( t )) = represents geometrica constrains. In our case the one possibiity exists for the transformation into independent coordinates, which, in this case, may be chosen as Cartesian coordinates of the center. Which is very suitabe because DAE mode is transformed to ODE mode []. It means that Lagrange s mutipies disappear and moreover we obtain more transparent reationship between centra working position and inputs - torques to the robot. hen the resuting mode of the robot is foowing: R MRy + R MRy = R g + R u () his mode can be generay rewritten in the state formua in foowing form: XD y () t = f ( X) + g( X) u() t () t = h X() t where the input variabes are torques of a drives. he state variabes are coordinates of center E (x, y) of movabe patform, its ange of winding ( ψ ) and their derivative. he output variabes are ony operationa coordinates seected from state variabes. (3) R -

3 he equations () and (3) are initia mathematica modes, which can be aready used according to requirements of design of contro. In order to use mode of the robot described by eq. (), we must provide the avaiabiity of independent coordinates i.e. coordinates of the center (x, y, ψ) of which number is equa to the number of degrees of freedom. his probem is soved by the direct kinematics, which recomputes joint-drive coordinates to Cartesian coordinates of movabe patform center. Before introduction of possibe approaches to the direct kinematics some pan of trajectory must be prepared. 3 PLANNING OF HE RAJECORY As opposed to cassica seria robots the trajectory panning for parae robots demands certain harmonization of robot movements. It is caused by interconnection of arms through the movabe patform vide Figure. Drive Drive 3 Drive 3 Drive (a) Drive Drive (b) Figure - Exampe of the cassica (a) and parae (b) types of the panar robots. he winding of a drives powering the parae robot gives simutaneousy position and winding of the movabe patform against cassica seria robots, where ange of winding is not stricty dependent on a drives, but it can be provided by ast drive (e.g. Figure.(a), Drive 3) in the chain. he trajectory is usuay given by technoogica requirements. And for contro, it must be time parameterized as discrete set of ordered pairs of time and Cartesian coordinates y [t, X = [ x yψ x ψ ]] t= k s at samping period s. In majority cases the whoe trajectory is composed of simpe segments, ike bisector or arc segments, which are simpy mathematicay described. In remaining cases the group of points is given and this eads on approximation ike smoothing probem. With using parametric interpoation or approximation curves e.g. ike Ferguson s cubic, Bezier s curves or B-spine curves, this probem is transformed on the former case. he genera requirement on panning of trajectory is a position of too and its ange of winding in certain time. Cartesian coordinates and required veocities give these. Note that the position and ange is identica with position and ange of winding of movabe patform. he design of the trajectory is based on simpe kinematics aws. At the first, the distanceength of segment and ange of winding, which is performed together, must be counted. Generay it is given by expressions s = ds ψ = ψ fina ψ inita ( a,b) s where ds is an eement of segment of the trajectory. R - 3

4 At using expressions for acceerations in a form: dv a =, dt dω α = (5 a,b) dt and their doube integration in frame of one segment we obtain expressions for orientationa working times: s t =, v inita + v fina ψ t = (6 a,b) ω + ω inita fina From them the higher, abeed as t fina, is chosen and rounded to the nearest vaue, which is mutipe of samping period s. hat provides sufficient time for performing of movement. Now the concrete parameterization can be made. For the smooth connecting and the accompishment of simutaneous movement and rotation of the movabe patform, the trajectory shoud have the first derivation continuous and smooth and at the same time the second derivation shoud be continuous and segmentay smooth curve with zero border vaues. In order to perform these requirements it is suitabe to prescribe the equation of acceeration as foowing: a = a + + (7) at at consecutivey veocity and position in the form: v inita = v + at + at at (8) s inita = v t + at + at at (9) with conditions of initia and fina state (as spoken above) in the form: for t = : v() = v inita ; s() = ; a() =, for t = t fina : v(t fina ) = v fina ; s(t fina ) = s fina ; a(t fina ) =. () From eq. (7) (8) (9) and conditions () the parameters a, a, a can be obtained. If we used the simiar equations for rotation α = α + α + α () t t 3 3 ω = ωinita + α t + αt + α t () 6 3 ψ = ωinitat + α t + αt + α t (3) and conditions in the form: for t = : ω () = ω inita; ψ() = ; α ()=. for t = t fina : ω (t fina ) = ω fina; ψ(t fina ) = ψ ; α (t fina ) =. () then we obtain aso the parameters for winding. R -

5 Now if we prepare time vector as foowing t =[, s, s ks] for integer k = fina (5) s and substitute it into eq. (7)-(9) and eq. ()-(3) we obtain consequence of positions, veocities and acceerations, which can be directy decomposed in directions x,y (Figure 3 and Figure ). he equations (7)-(9) and eq. ()-(3) can be used for computation of the other geometrica parameterization [5]. hen required ordered pairs of time and coordinates with their derivatives are obtained. his time parameterization of the trajectory can be aready used for rea - time contro, even for the design and testing of the direct kinematics, which is discussed in the foowing section. y [m] x [m] Figure 3 - Exampe of one trajectory, composed of bisector and arc segments. x[m] y[m] ψ [ ] vx[m s ] vy[m s - ] ωψ [rad s - ] ax[m s ] ay[m s - ] αψ [rad s ] Figure - Kinematic component characterizations of the trajectory from Figure 3. (positions [x, y, ψ], veocities and acceerations). R - 5

6 DIREC KINEMAICS he coordinates appearing in the branch of robots and manipuators may be divided into drive coordinates q, operationa x and other anciary coordinates q. A these coordinates are either independent (their number equas number of degrees of freedom) or dependent. Among them there are reations generay expressed by system of noninear equations: f x, q, q ) = (6) ( Direct kinematics soves the probem of recomputation the drives coordinates q on operationa, in our case (Figure ), independent coordinates x. In comparison with the cassica robots, where it is not difficut, the direct kinematics of the parae robots especiay redundanty actuated is not simpe task. hen, we search the function x = f ( q ), which is not unfortunatey in case of parae robots anayticay soved. We have severa possibiities, how to sove this. Either we can use cassica numerica soution or engineering soution in the form of contro task.. Numerica soution Now we briefy describe usua numerica soution with the Newton s method, which is suitabe. he method is based on the ayor expansion to the first order in the neighbourhood of initia ( ) ( ) ( vaue Z = [ x ; q ) ], which can be equa to desired vaues or to vaues from the previous step. ( k ) k f ( Z, q) ( k ) f, + Z (7) ( ) ( Z q ) = f ( Z, q ) = Z From this expansion we obtain a system of inear equations for Z (k) f ( k ) ( Z, q ) ( k ) ( k ) Z Z then k + iteration is = f Z ( Z, q ) ( k +) ( k ) ( k ) = Z + Z he resut (9) we can substitute into eq. (8) and repeat procedure unti Z difference given before, which is aready not critica for contro. Consecutivey the veocities and acceerations are given by d dt f Z d dt ( f ( Z, q) = ) ( Z, q ) f ( Z, q ) ZD + q qd = ZD ( k ) (8) (9) ε, where ε is such ZD + Φ qd = ZD Z q () Φ ( Φ ZD + Φ qd = ) Z q D ZD + Φ ZDD + ΦD qd + Φ qdd = ZD Z Z q q () Φ With using the previous resuts of eq. (9) for eq. () and eq. (9) and () for eq. (), the eq. () and () represent systems of inear agebraic equations. From them the operationa coordinates arise. R - 6

7 . Soution in the form of contro task As has been mentioned, the direct kinematics of the parae robots is more compicated and it has not direct anaytica soution as at cassica kinematic structures, where the direct kinematics is simpe and conversey there is a probem with kinematics inversion there. When we consider the previous numerica soution, we can see, that after derivation of system equations (6), we obtain inear reations. his fact we can use. If we have possibiity to extract the reation ony between independent operationa coordinates x and drive coordinates q : f x, q ) = q = f ( ) () ( x and derivate it according to time we obtain system of inear differentia kinematic equations qd df = xd dx qd J xd = (3) where J is Jacobian. hen eq. (3) makes the basis of the foowing approach. Suppose that desired vaues x d of the trajectory are avaiabe and the same may be said about initia conditions on position and ange of winding. By considering eq. (3) with reguar square matrix J the operationa coordinates can be obtained via simpe inversion ( ) - x = J q () In the case, when parae robot is redundant (our case Figure ) the Jacobian has more rows than coumns and it can t be inverted. We can use eft pseudo-inverse. ( J J ) - J q x = (5) he Jacobian J is a function of operationa coordinates x d and if difference x d -x is beow a given toerated threshod then the eq. () or (3) can be integrated. For rea impementation, it is mosty needfu to rewrite these equations to discrete form. Note the Jacobian at the certain instant is ony static reation, which is not directy dependent on time. So we can write: x t x t + xd t ( k ) = ( k ) ( k ) s ( tk ) = x( tk ) + J ( xd ( t k )) q( tk ) s where ( ) x D (6) J x d is either simpe inversion eq. () or eft pseudo-inverse eq. (5) of Jacobian according to type of the robot. However in practice the differences x d -x can be higher than a given toerated threshod or we need to move with the robot freey without the knowedge of desired trajectory. hen in numerica impementation of eq. (6), computation of operationa veocities is obtained by using the inverse of the Jacobian evauated at the previous instant of time ( t ) x( t ) + J ( x( t )) q ( t ) s k = k k k x D (7) R - 7

8 Eq. (7) does not satisfy eq. (), eq. (5) respectivey. his inconvenience can be overcome by approach to such scheme, which takes into account the difference between actua topica measured joint-drive coordinates and recomputed joint-drive coordinates from computed (estimated) operationa coordinates. Let ( x) e = qm q (8) be the expression of such difference. Consider the time derivative of eq. (8) = q q x (9) e M ( ) which, according to the differentia kinematics eq. (3), can be rewritten as e = q M J ( x)x (3) his equation eads to a feedback scheme of the direct kinematics and reation eq. (3) between operationa veocity xd and derivative of difference ec gives a differentia equation, which describes difference evoution over time. Nonetheess, it is necessary to choose a reation between xd and e that ensures convergence of the difference to zero. Assume the choice! ( x) x = K( q q ( x) ) = Ke e = q M J M (3) which eads to the equivaent inear system e + Ke = (3) If K is a positive definite (usuay diagona) matrix, the system eq. (3) is asymptoticay stabe. hen we can rewrite eq. (3) to the form J and consecutivey where ( x) ( x) x = qd + K( q q ( x) ) D (33) M M ( x) ( q + K( q q ( x) )) xd = J D (3) M M J has a simiar meaning, as in previous section, either simpe inversion or eft pseudoinverse accordingy to type of the robot. Finay, the discrete form of eq. (3) is given by expression x ( t ) x( t ) + J ( x( t )) [ q ( t ) + K[ q ( t ) q ( x ( t ))]] s k D (35) = k k- M k M k k he difference e tends to zero aong measured joint-drive coordinates q M with convergence rate that shoud be provided by K <. he bock scheme corresponding to the feedback direct kinematics agorithm eq. (35) for redundant case is in Figure 6. R - 8

9 Input q M [m ] z Unit deay + K Gain + qd M [m ] /s Gain Ke + q D M + Ke + ( J J ) J [ n m ] Left pseudo-inverze s z - Product Discrete -ime Integrator J [ m n ] m > n Output Output xd computed [n ] x computed [n ] Feedback Inverse kinematics q (x) x Computation of Jacobian Figure 6 - Feedback direct kinematics. 5 EVALUAION OF PRESENED APPROACHES In this section we focus on the ast approach, on feedback agorithm of direct kinematics. he former approach (Newton s method) is not bad, but it is sower. It is caused by iteration character of the agorithm. For introduction, the simpe trajectory, composed of spira and arc segment, was chosen Figure 7. y [m] x [m] Figure 7 - Desired vaues of operationa coordinates Cartesian coordinates: x, y, ψ. ψ [ ] he robot (Figure ) begins from center of the workspace and tracking the trajectory with sow increase of sin trend of ange of winding of movabe patform. x -5 [m] x -3 [ ] difference of x - - difference of y difference of ψ -6 3 Figure 8 - Simuation time history of differences. x -3 [m] difference of y 8 6 [ ] - difference of x difference of ψ - 3 Figure 9 - Rea time history of differences. R - 9

10 he Figure 8 and Figure 9 show the time history of differences (errors) between desired vaues x d and computed (estimated) vaues x. For rea time process, the samping s equaed.s and for simuation samping s was.s. he positive diagona matrix was chosen with diagona eement k ij = Figure - Rea-time comparison of desired (soid) and computed (dotted) trajectory. For rea time test the simpe proportiona controer was used and moreover the robot was not ideay caibrated. hat is why the difference between desired x d and computed x operationa coordinates is greater than in simuation case, where the direct kinematic agorithms were tested directy without contro. 6 CONCLUSION Presented approaches to the direct kinematics are suitabe for simuation (Newton s method) and mainy for rea time using (Newton s method, feedback agorithm). hey were successfuy tested and shown in this paper. References.. ORDYS, A., CLARKE, D. A state space description for GPC controers. IN. J. Systems SCI., 993, vo., no.9, pp BÖHM, J., BELDA, K., VALÁŠEK, M. he direct kinematics for Path Contro of Redundant Parae Robots, In.: Advances in System science: Measurement, Circuits and Contro, 5th Word Conference on Systems, Crete - Greece, WSES,, pp , ISBN: SEJSKAL, V., VALÁŠEK, M. Kinematics and dynamics of machinery. New York: Marce Dekker, Inc., 996, 9 pp.. VALÁŠEK, M. Dynamic time parameterisation of manipuator trajectories, Kybernetika, 987, Vo. 3, Num., pp SCIAVICCO, L., SICILIANO, B. Modeing and contro of robot manipuators, McGraw-Hi, 996. Acknowedgments his research has been supported by CU grant No. 3, GAČR grant No. /9979. R -

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